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We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\\le d\\le 36$), the smallest prime $p\\equiv c\\pmod d$ with $p\\ge(2dn-c)/(d-1)$ is the least positive integer $m$ with $2r(d)k(dk-c)\\ (k=1,\\ldots,n)$ pairwise distinct modulo $m$, where $r(d)$ is the radical of $d$. We also conjecture that for any integer $n>4$ the least positive integer $m$ such that $|\\{k(k-1)/2\\ \\mbox{mod}\\ m:\\ k=1,\\ldots,n\\}|= |\\{k(k-1)/2\\ \\mbox{mod}\\ m+2:\\ k=1,\\ldots,n\\}|=n$ is the least prime $p\\ge 2n-1$ with $p+2$ also prim"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1304.5988","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-22T15:39:13Z","cross_cats_sorted":[],"title_canon_sha256":"3beb8c2f50c0ff3b9f04d927e5e4a48c982de02a5dafb8acbfa17647c27c93ac","abstract_canon_sha256":"40967f4587826c9eac271870666cdc77b279fef3f19ea129474263536e35f27b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:29:32.160549Z","signature_b64":"A6fIKGYFCz2V8OH4LfIFVuvo4szayMf6CpileRuLPKA0yX0QREy7he3MhYGQ81xGIhpKQTCHS7WUrHz9RvIoAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3415443392cb622bb3ea802fb1ca7c027c7502cbf8419c516a9def763d2beff2","last_reissued_at":"2026-05-18T01:29:32.159815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:29:32.159815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The least modulus for which consecutive polynomial values are distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2013-04-22T15:39:13Z","abstract_excerpt":"Let $d\\ge4$ and $c\\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\\le d\\le 36$), the smallest prime $p\\equiv c\\pmod d$ with $p\\ge(2dn-c)/(d-1)$ is the least positive integer $m$ with $2r(d)k(dk-c)\\ (k=1,\\ldots,n)$ pairwise distinct modulo $m$, where $r(d)$ is the radical of $d$. We also conjecture that for any integer $n>4$ the least positive integer $m$ such that $|\\{k(k-1)/2\\ \\mbox{mod}\\ m:\\ k=1,\\ldots,n\\}|= |\\{k(k-1)/2\\ \\mbox{mod}\\ m+2:\\ k=1,\\ldots,n\\}|=n$ is the least prime $p\\ge 2n-1$ with $p+2$ also prim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5988","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1304.5988","created_at":"2026-05-18T01:29:32.159922+00:00"},{"alias_kind":"arxiv_version","alias_value":"1304.5988v5","created_at":"2026-05-18T01:29:32.159922+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.5988","created_at":"2026-05-18T01:29:32.159922+00:00"},{"alias_kind":"pith_short_12","alias_value":"GQKUIM4SZNRC","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"GQKUIM4SZNRCXM7K","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"GQKUIM4S","created_at":"2026-05-18T12:27:45.050594+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ","json":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ.json","graph_json":"https://pith.science/api/pith-number/GQKUIM4SZNRCXM7KQAX3DST4AJ/graph.json","events_json":"https://pith.science/api/pith-number/GQKUIM4SZNRCXM7KQAX3DST4AJ/events.json","paper":"https://pith.science/paper/GQKUIM4S"},"agent_actions":{"view_html":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ","download_json":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ.json","view_paper":"https://pith.science/paper/GQKUIM4S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1304.5988&json=true","fetch_graph":"https://pith.science/api/pith-number/GQKUIM4SZNRCXM7KQAX3DST4AJ/graph.json","fetch_events":"https://pith.science/api/pith-number/GQKUIM4SZNRCXM7KQAX3DST4AJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ/action/storage_attestation","attest_author":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ/action/author_attestation","sign_citation":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ/action/citation_signature","submit_replication":"https://pith.science/pith/GQKUIM4SZNRCXM7KQAX3DST4AJ/action/replication_record"}},"created_at":"2026-05-18T01:29:32.159922+00:00","updated_at":"2026-05-18T01:29:32.159922+00:00"}